A natural model and a parallel algorithm for approximately solving the maximum weighted independent set problem

A natural model and a parallel algorithm for approximately solving the maximum weighted independent set problem
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  Chaoc, Solams & Fracrals Vol. 5, No. 5. pp. 139-746. 1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved C960-0779/9S$9SP + .(K) 0960-0779(94)00175-S A Natural Model and a Parallel Algorithm for Approximately Solving the Maximum Weighted Independent Set Problem MOHAMED AFIF LAMSADE, Universite Paris-Dauphine, Place du Mar&ha1 De Lattre de Tassigny, 75775 Paris, Cedex 16, France ARISTIDIS LIKAS Department of Electrical and Computer Engineering, National Technical University of Athens, 157 73 Zographou, Athens, Greece and VANGELIS TH. PASCHOS LAMSADE, Universite Paris-Dauphine, Place du Mar&ha1 De Lattre de Tassigny, 75775 Paris, Cedex 16, France (Received 17 June 1994) Abstract-A dynamical model based upon a physical metaphor is described, and a parallel algorithm inspired from the model is developed for approximately solving maximum weight independent set problem. Our model treats an independent set as an attraction game, where vertices of the graph are considered as still bodies and edges as cells attracted by the still bodies corresponding to its extremities. In addition, we discuss how, by using an analogous model, an approximation algorithm can be developed for the minimum set covering problem. 1. INTRODUCTION Consider a graph G = (V, E) of order n. An independent set is a subset of S c V such that there is no pair of nodes in S linked by an edge in G and the maximum independent set problem (IS) is to find an independent set of maximum size. This problem admits a natural generalization, in the case where we associate positive weights (costs) wi, i = 1, . . .) iz, to the vertices of its instance, and then the objective becomes to maximize the sum of the weights of the vertices of an independent set. (In what follows, we denote by WIS the weighted version of the maximum independent set problem.) WIS constitutes one of the most famous NP-complete problems and, in addition, even its constant approximability in polynomial time is NP-complete also [l]. For this reason, a number of sequential [2-41 or parallel [5,6] heuristics have been developed for IS (both the weighted and unweighted cases). In this paper inspired by [7] where a dynamical system and a parallel algorithm have been developed for the minimum weighted vertex covering problem, we propose a model for WIS. This model, based upon a magnetism metaphor, is described by a dynamical system giving rise to a parallel algorithm approximately solving WE. Moreover, it seems 739  740 M. AFIF et al. that the proposed model is well-adapted to the problem since, as is proved below, the dynamical system converges to an equilibrium state corresponding to a feasible WE solution. Moreover, we show that, by extending the proposed approach, it is possible to define an analogous model for approximately solving the minimum set covering problem. Given a collection 9’ (191 = n) of subsets of a finite set C (/Cl = m), a cover is a subcollection 7 & Y such that Us,Ey” S, = C, and the minimum set covering problem (SC) is to find a cover of minimum size. By associating weights w,, i = 1, . . . , II, to the sets of Y. we obtain WSC (weighted set covering), a generalization of SC. The objective of WSC is to minimize the total weight of a set covering, this weight being the sum of the weights of the sets in the covering. In [8], Hifi et al. propose a Boltzmann machine architecture, treating SC as a particular instance of the minimum vertex covering problem. In this paper, we propose a natural model for WSC fairly similar to the one for WIS, and we prove that the dynamical system describing the WSC model reaches stability; however, as we can see, this model has 1:he drawback of not always providing feasible solutions. In what follows, given an undirected graph G = (V, E) of order ~1, we denote by r( ri,) and w, the neighbour set and the weight of each vertex u, (u, E V); also, we denote by h, the cardinality Ir(~,)l (the degree of u,). 2. THE DYNAMICAL SYSTEM AND THE DEDUCED PARALLEL APPROXIMATION ALGORITHM FOR THE WEIGHTED INDEPENDENT SET The natural metaphor we have used to model and solve WIS is the following. Consider the vertex set V as a set of still bodies placed on a bi-dimensional space, and the edge set E as a set of cells located between pairs of still bodies, each cell being likely to be attracted by one of the still bodies that lie on the extremities of the corresponding edge.” Once a vertex’ succeeds to attract one of the edges incident to it, then it immediately attracts all of its incident edges and wins in the competition over its neighbours, this victory entailing the inclusion of the winner in the WIS solution and the corresponding exclusion of its neighbours. A potential U,(t) is associated to every vertex ui E V, for any instant t. The motion of a cell cU (corresponding to the edge UiU,), located between ui and Uj, is described at any instant t by a function xii(t). It holds that .x;,(t) -+ ~0 whenever cti is located close to u,, xl,(f) -+ ---CO henever cij is located close to Ui, while whenever c;j lies in the middle of the distance between ui and Uj, xii(t) = 0. Finally, since G is undirected, we consider that the cell cji is identical to the cell Cij. Cell ci, moves towards the body argmax,,t,,EE{ L’i(t)t uj(t)} with velocity Ir-ij t) = Uj t) - Ui(t), vivj E E. (1) Moreover, we define the following transformation r(l) which constraints the values of xlj(t) within the interval [-1, + 11: rij(t) = tanh (r~q(t)), u,u, E E (2) where y is a positive constant (parameter of the system) that allows the adjustment of the hyperbolic tangent’s slope. Based on the above transformation, we have rii(t) + +1 - *Of course, we suppose that the cells are initially located at positions equidistant from the associated bodies: moreover, these distances are finite. *From here on we shall treat he terms till body and vertex as equivalent; we do the same for the terms cell and edge, respectively.  Maximum weighted independent set problem 741 whenever c,, is located close to “j, r~j(t) + - 1 whenever cli is located close to uI, while whenever cj, lies in equal distance from U, and ui, rJt) = 0. Let us note that, since hyperbolic tangent function, tanh, is odd, we have yi,( t) = - rji(t) for every instant t. It remains now to define the potential of each vertex. This is defined in such a way that U,(t) E [ - w, , + wJ, ui E V. Let us remark that a natural definition of such potential would be u,(t) = (wl/Wc o EV~,i( ), but whenever G contains isolated vertices then the potential of these vertices wou d be undefined; so, we set U;(t) &[( o,~L,,~j~(f)) l]O. E v. (3) It is easy to see that in the case where 6, = 0, Ui(t) = Wi. The system of differential equations induced by expressions (l)-(3) describes the evolution of the dynamical system over time. Based on these expressions, we can define a WIS solution S in the following way: s = {Ui E v: ui = pir ui(t) = w,}. (4) Proposition 1. The set S defined by expression (4) is feasible for WIS. Proof. Let US suppose that there exists an edge ViUj E E such that {Ui, Uj} c S. Since ui E S, Ui = Wi and from expression (3), we get C U,Er(Uijr,i(t) = 6i; this last expression implies that, for all u,uk E E, rki = 1; SO, rji = 1. If we follow the same arguments for the fact that u, E S, we can also find that ri, = 1, which is a contradiction since, by the definition of the model, ri,(t) = - r,i(t) for every t. n Let us note that the quantity xi,(t) can be written as xii(t) = xc(to) + (:,l(dxi,(s)/ds)ds. So, by discretization, we obtain ’ dx. (s) X*j(t + dt) = Xii(fo) + dt C II, s=t"(dt) ds (5) By means of expressions (l)-(5), we can specify the WIS Algorithm 1. It must be noted that only quantities computed at instant t are necessary for the computations at instant f f dt . Therefore, Algorithm 1 can very easily be implemented on a parallel machine. It remains now to show that the described dynamical system (and, consequently, Algorithm 1 based upon this system) converges to an equilibrium state. To do this, it is sufficient to show that there is a function E(t) constituting a Lyapunov function for the system or, in other words, admitting the property i < 0. Proposition 2. The function E(t) = k c C’ij(l)&xrik(t) + c c &r,(t) u, 0, I Ok I’, 0, 1 begin fix constants y, E, E; fix a time-step dt; t+--0  742 M. AFIF et al. for all UiUj E E do xq(0) + E; rii(0) t tanh (yxii(0)) od repeat for for alUiE V do vi t) + Wiai+ l)[ Co,t~ ~:,)rji t)) f 11 od all UiU, E E do x,(t + dt) +xV(~) + dt(Uj(t) - Ui(t)); rii(t -t dt) ctanh(yxU(t + dt)); ttt+dt od until lrij t) - l/ d E S = {Ui E VI lim,,, ui(t) = Wi} end. Algorithm 1 constitutes a Lyapunov function for the system described by expressions (l)-(3). Proof. Let US recall that rii(t) = -rji(t), for every t; also Xi,(t) = -Xii(t) and, by model definition, Cwd6t + l)(C i-ii(t) = -iji t), drij/dXij = drji/dxjl. It holds that dE/dr, = u,ukeErik 1) = Ui (by expression (3)), independently of Uj. Moreover, E(t) = cII,cU ,jci(dE/dr, - dE/drji)(dr,/dx,)~i-,(t), this expression yielding k(t) = --Co,Co,,j<i(dj - ui)2(drv/dXq) c 0. n 3. EXPERIMENTAL RESULTS We have tested Algorithm 1 considering two performance criteria: execution time (in seconds) and approximation ratio. j In fact, concerning the latter criterion, we have estimated the quantity p = (~Ipl)/lSl, where pz is the approximation ratio of the algorithm on an IS instance I and 1411 s the number of the produced graphs. Moreover, in order to have some comparative performance information, we have implemented the natural greedy algorithm 2 for IS, and we have tested it following the same performance criteria. Finally, in order to find the optimal solutions of the tested instances, we have implemented a branch-and-bound method adapted to IS. begin St0 repeat u, + argmax u,E~ w,/6j) ; S+S U {Ui}; v + v\({“~) u r(“i)) until V = 0 end. Algorithm 2 - *In polynomial approximation theory, a usual criterion of the performance of a heuristic on an instance of an h’P-complete problem is the ratio ‘size (value in the weighted case) of the solution provided by the heuristic over site (or value) of the optimal solution of the instance’.  Maximum weighted independent set problem 743 We have generated randomly 161 graphs, 89 among them being regular and 72 irregular. The orders of the generated graphs vary between 15 and 100, and the numbers of their edges between 20 and 200. For the regular graphs, the degrees of the vertices vary between 2 and 7, while for the irregular ones the degrees vary between 1 and 7. Both methods have been executed twice on each instance: one time for the unweighted case and one time by considering integer weights on the vertices, these weights being randomly chosen from the set (1, . . . 30). The tests have been executed on a 33 MHz 486 PC; the execution times for Algorithm 1 concern this sequential implementation. 3.1. A summary of the performance of the two algorithms We present in this section a summary of the performance (concerning average case ratios and average execution times) of the proposed approach (Algorithm 1) and the greedy heuristic (Algorithm 2), respectively. Table 1 summarizes the performance of the two heuristics for the case where all of the weights are equal to 1 (for this case, the selection performed at each step of Algorithm 2 becomes a simple minimum degree vertex choice), while in Table 2, weighted cases are considered. Since Algorithm 1 uses the parameters E, y and dt, the performance of the algorithm, concerning both approximation ratios and execution times, depends on the values of these parameters. In this section, we present the parameter values allowing us to obtain good approximation results in reasonable execution times; in Section 3.2, we present results from experiments concerning the influence of these three parameters upon the performance of Algorithm 1. As one can see from Tables 1 and 2, the experimental behaviour of the conceived model is quite satisfactory and superior to the one of the greedy algorithm, in what concerns the average case approximation ratio. Of course, concerning the corresponding execution times, we note here that we have executed Algorithm 1 sequentially. However, since it is easily parallelizable, its parallel execution time is expected to be much smaller and comparable to the one of the greedy algorithm. 3.2. Influence of the parameters E, y and dt on the behaviour of Algorithm I The parameters E (computation’s precision), y (slope of the function tanh) and dt (computation step) used by Algorithm 1 play an important role on both the approximation ratio and the execution time of the algorithm. So, in Tables 3-5, we present experimental Table 1. Unweighted IS approximation performances of the two heuristics; Algorithm 1 has been executed with E = 10m8, dt = 10m3 and y = 250 Method P % Optimal solutions Average execution time Algorithm 1 0.947 57.14 10.8 Algorithm 2 0.918 41.61 0.77 Table 2. Weighted IS approximation performances of the two heuristics; Algorithm 1 has been executed with E = lo-*, dt = 10m4 and y = 200 Method P % Optimal solutions Average execution time Algorithm 1 0.963 39.75 12.8 Algorithm 2 0.959 36.65 1.02
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